| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lmod1.m |
⊢ 𝑀 = ( { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑦 ∈ { 𝐼 } ↦ 𝑦 ) ⟩ } ) |
| 2 |
|
eqid |
⊢ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } = { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } |
| 3 |
2
|
grp1 |
⊢ ( 𝐼 ∈ 𝑉 → { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ∈ Grp ) |
| 4 |
|
fvex |
⊢ ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ∈ V |
| 5 |
|
snex |
⊢ { 𝐼 } ∈ V |
| 6 |
2
|
grpbase |
⊢ ( { 𝐼 } ∈ V → { 𝐼 } = ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ) |
| 7 |
5 6
|
ax-mp |
⊢ { 𝐼 } = ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) |
| 8 |
7
|
opeq2i |
⊢ ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ = ⟨ ( Base ‘ ndx ) , ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ |
| 9 |
|
tpeq1 |
⊢ ( ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ = ⟨ ( Base ‘ ndx ) , ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ → { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } = { ⟨ ( Base ‘ ndx ) , ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ) |
| 10 |
8 9
|
ax-mp |
⊢ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } = { ⟨ ( Base ‘ ndx ) , ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } |
| 11 |
10
|
uneq1i |
⊢ ( { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑦 ∈ { 𝐼 } ↦ 𝑦 ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑦 ∈ { 𝐼 } ↦ 𝑦 ) ⟩ } ) |
| 12 |
1 11
|
eqtri |
⊢ 𝑀 = ( { ⟨ ( Base ‘ ndx ) , ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑦 ∈ { 𝐼 } ↦ 𝑦 ) ⟩ } ) |
| 13 |
12
|
lmodbase |
⊢ ( ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ∈ V → ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) = ( Base ‘ 𝑀 ) ) |
| 14 |
4 13
|
ax-mp |
⊢ ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) = ( Base ‘ 𝑀 ) |
| 15 |
14
|
eqcomi |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) |
| 16 |
|
fvex |
⊢ ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ∈ V |
| 17 |
|
snex |
⊢ { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ∈ V |
| 18 |
2
|
grpplusg |
⊢ ( { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ∈ V → { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } = ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ) |
| 19 |
17 18
|
ax-mp |
⊢ { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } = ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) |
| 20 |
19
|
opeq2i |
⊢ ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ = ⟨ ( +g ‘ ndx ) , ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ |
| 21 |
|
tpeq2 |
⊢ ( ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ = ⟨ ( +g ‘ ndx ) , ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ → { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } = { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ) |
| 22 |
20 21
|
ax-mp |
⊢ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } = { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } |
| 23 |
22
|
uneq1i |
⊢ ( { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑦 ∈ { 𝐼 } ↦ 𝑦 ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑦 ∈ { 𝐼 } ↦ 𝑦 ) ⟩ } ) |
| 24 |
1 23
|
eqtri |
⊢ 𝑀 = ( { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑦 ∈ { 𝐼 } ↦ 𝑦 ) ⟩ } ) |
| 25 |
24
|
lmodplusg |
⊢ ( ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) ∈ V → ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) = ( +g ‘ 𝑀 ) ) |
| 26 |
16 25
|
ax-mp |
⊢ ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) = ( +g ‘ 𝑀 ) |
| 27 |
26
|
eqcomi |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ) |
| 28 |
15 27
|
grpprop |
⊢ ( 𝑀 ∈ Grp ↔ { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ } ∈ Grp ) |
| 29 |
3 28
|
sylibr |
⊢ ( 𝐼 ∈ 𝑉 → 𝑀 ∈ Grp ) |
| 30 |
29
|
adantr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝑀 ∈ Grp ) |
| 31 |
1
|
lmodsca |
⊢ ( 𝑅 ∈ Ring → 𝑅 = ( Scalar ‘ 𝑀 ) ) |
| 32 |
31
|
eqcomd |
⊢ ( 𝑅 ∈ Ring → ( Scalar ‘ 𝑀 ) = 𝑅 ) |
| 33 |
32
|
adantl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( Scalar ‘ 𝑀 ) = 𝑅 ) |
| 34 |
|
simpr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝑅 ∈ Ring ) |
| 35 |
33 34
|
eqeltrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( Scalar ‘ 𝑀 ) ∈ Ring ) |
| 36 |
33
|
fveq2d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( Base ‘ ( Scalar ‘ 𝑀 ) ) = ( Base ‘ 𝑅 ) ) |
| 37 |
36
|
eleq2d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↔ 𝑞 ∈ ( Base ‘ 𝑅 ) ) ) |
| 38 |
36
|
eleq2d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↔ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) |
| 39 |
37 38
|
anbi12d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ( 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) ↔ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) ) |
| 40 |
|
simpll |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → 𝐼 ∈ 𝑉 ) |
| 41 |
|
simplr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑅 ∈ Ring ) |
| 42 |
|
simprr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑟 ∈ ( Base ‘ 𝑅 ) ) |
| 43 |
40 41 42
|
3jca |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) |
| 44 |
1
|
lmod1lem1 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ) |
| 45 |
43 44
|
syl |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ) |
| 46 |
1
|
lmod1lem2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) |
| 47 |
43 46
|
syl |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) |
| 48 |
1
|
lmod1lem3 |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) |
| 49 |
45 47 48
|
3jca |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ) |
| 50 |
1
|
lmod1lem4 |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) |
| 51 |
1
|
lmod1lem5 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) |
| 52 |
51
|
adantr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) |
| 53 |
49 50 52
|
jca32 |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) |
| 54 |
53
|
ex |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ( 𝑞 ∈ ( Base ‘ 𝑅 ) ∧ 𝑟 ∈ ( Base ‘ 𝑅 ) ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) ) |
| 55 |
39 54
|
sylbid |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ( 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) ) |
| 56 |
55
|
ralrimivv |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ∀ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) |
| 57 |
|
oveq2 |
⊢ ( 𝑥 = 𝐼 → ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) = ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) |
| 58 |
57
|
oveq2d |
⊢ ( 𝑥 = 𝐼 → ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) ) |
| 59 |
|
oveq2 |
⊢ ( 𝑥 = 𝐼 → ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) |
| 60 |
59
|
oveq2d |
⊢ ( 𝑥 = 𝐼 → ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) |
| 61 |
58 60
|
eqeq12d |
⊢ ( 𝑥 = 𝐼 → ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ↔ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ) |
| 62 |
61
|
3anbi2d |
⊢ ( 𝑥 = 𝐼 → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ↔ ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ) ) |
| 63 |
62
|
anbi1d |
⊢ ( 𝑥 = 𝐼 → ( ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ↔ ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ) ) |
| 64 |
63
|
ralbidv |
⊢ ( 𝑥 = 𝐼 → ( ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ↔ ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ) ) |
| 65 |
64
|
ralsng |
⊢ ( 𝐼 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐼 } ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ↔ ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ) ) |
| 66 |
65
|
adantr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ∀ 𝑥 ∈ { 𝐼 } ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ↔ ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ) ) |
| 67 |
|
oveq2 |
⊢ ( 𝑤 = 𝐼 → ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) |
| 68 |
67
|
eleq1d |
⊢ ( 𝑤 = 𝐼 → ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ↔ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ) ) |
| 69 |
|
oveq1 |
⊢ ( 𝑤 = 𝐼 → ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) = ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) |
| 70 |
69
|
oveq2d |
⊢ ( 𝑤 = 𝐼 → ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) ) |
| 71 |
67
|
oveq1d |
⊢ ( 𝑤 = 𝐼 → ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) |
| 72 |
70 71
|
eqeq12d |
⊢ ( 𝑤 = 𝐼 → ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ↔ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ) |
| 73 |
|
oveq2 |
⊢ ( 𝑤 = 𝐼 → ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) |
| 74 |
|
oveq2 |
⊢ ( 𝑤 = 𝐼 → ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) |
| 75 |
74 67
|
oveq12d |
⊢ ( 𝑤 = 𝐼 → ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) |
| 76 |
73 75
|
eqeq12d |
⊢ ( 𝑤 = 𝐼 → ( ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ↔ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ) |
| 77 |
68 72 76
|
3anbi123d |
⊢ ( 𝑤 = 𝐼 → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ↔ ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ) ) |
| 78 |
|
oveq2 |
⊢ ( 𝑤 = 𝐼 → ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) |
| 79 |
67
|
oveq2d |
⊢ ( 𝑤 = 𝐼 → ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) |
| 80 |
78 79
|
eqeq12d |
⊢ ( 𝑤 = 𝐼 → ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ↔ ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ) |
| 81 |
|
oveq2 |
⊢ ( 𝑤 = 𝐼 → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) |
| 82 |
|
id |
⊢ ( 𝑤 = 𝐼 → 𝑤 = 𝐼 ) |
| 83 |
81 82
|
eqeq12d |
⊢ ( 𝑤 = 𝐼 → ( ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ↔ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) |
| 84 |
80 83
|
anbi12d |
⊢ ( 𝑤 = 𝐼 → ( ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ↔ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) |
| 85 |
77 84
|
anbi12d |
⊢ ( 𝑤 = 𝐼 → ( ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ↔ ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) ) |
| 86 |
85
|
ralsng |
⊢ ( 𝐼 ∈ 𝑉 → ( ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ↔ ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) ) |
| 87 |
86
|
adantr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ↔ ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) ) |
| 88 |
66 87
|
bitrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ∀ 𝑥 ∈ { 𝐼 } ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ↔ ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) ) |
| 89 |
88
|
2ralbidv |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ∀ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∀ 𝑥 ∈ { 𝐼 } ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ↔ ∀ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝐼 ( +g ‘ 𝑀 ) 𝐼 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝐼 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝐼 ) = 𝐼 ) ) ) ) |
| 90 |
56 89
|
mpbird |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ∀ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∀ 𝑥 ∈ { 𝐼 } ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ) |
| 91 |
1
|
lmodbase |
⊢ ( { 𝐼 } ∈ V → { 𝐼 } = ( Base ‘ 𝑀 ) ) |
| 92 |
5 91
|
ax-mp |
⊢ { 𝐼 } = ( Base ‘ 𝑀 ) |
| 93 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
| 94 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑀 ) = ( ·𝑠 ‘ 𝑀 ) |
| 95 |
|
eqid |
⊢ ( Scalar ‘ 𝑀 ) = ( Scalar ‘ 𝑀 ) |
| 96 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑀 ) ) = ( Base ‘ ( Scalar ‘ 𝑀 ) ) |
| 97 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑀 ) ) = ( +g ‘ ( Scalar ‘ 𝑀 ) ) |
| 98 |
|
eqid |
⊢ ( .r ‘ ( Scalar ‘ 𝑀 ) ) = ( .r ‘ ( Scalar ‘ 𝑀 ) ) |
| 99 |
|
eqid |
⊢ ( 1r ‘ ( Scalar ‘ 𝑀 ) ) = ( 1r ‘ ( Scalar ‘ 𝑀 ) ) |
| 100 |
92 93 94 95 96 97 98 99
|
islmod |
⊢ ( 𝑀 ∈ LMod ↔ ( 𝑀 ∈ Grp ∧ ( Scalar ‘ 𝑀 ) ∈ Ring ∧ ∀ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∀ 𝑥 ∈ { 𝐼 } ∀ 𝑤 ∈ { 𝐼 } ( ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ∈ { 𝐼 } ∧ ( 𝑟 ( ·𝑠 ‘ 𝑀 ) ( 𝑤 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ( +g ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑀 ) ( 𝑟 ( ·𝑠 ‘ 𝑀 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑀 ) 𝑤 ) = 𝑤 ) ) ) ) |
| 101 |
30 35 90 100
|
syl3anbrc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝑀 ∈ LMod ) |